1. 2.5 Using Linear Models
Month
Temp 1 2 3 4
69 º F
70 º F
75 º F
78 º F

2. 2.5 Using Linear Models Scatter Plot –
A graph that relates two sets of data by plotting the data as
ordered pairs

3. 2.5 Using Linear Models
A scatter plot can be used to determine the strength of the relation or the correlation between data sets.
The closer the data points fall along a line with a positive slope,
The stronger the linear relationship, and
the stronger the positive correlation

4. 2.5 Using Linear Models STRONG WEAK POSITIVE CORRELATION
Describe the correlation shown in each graph.
STRONG
POSITIVE CORRELATION
WEAK
POSITIVE CORRELATION

5. 2.5 Using Linear Models
STRONG
NEGATIVE
CORRELATION
NO CORRELATION

6. 2.5 Using Linear Models
Is there a positive, negative, or no correlation between the 2 quantities? If there is a positive or negative correlation, is it strong or weak?

7. A person’s age and his height
2.5 Using Linear Models
Age (in years)
Height (in feet)
A person’s age and his height
POSITIVE
STRONG

8. A person’s age and the number of cartoons he watches
2.5 Using Linear Models
A person’s age and the number of cartoons he watches
NEGATIVE
WEAK

9. 2.5 Using Linear Models
The table shows the median home prices in New Jersey. An equation is given that models the relationship between time and home prices. Use the equation to predict the median home price in

10. 2.5 Using Linear Models y = 2061x + 47,100
YEAR
MEDIAN PRICE ($)
1940
47,100
1950
63,100
1960
76,900
1970
89,900
1980
119,200
1990
207,400
2000
170,800
y = 2061x + 47,100
where x is the number of years
since 1940
and y is the price

11. 2.5 Using Linear Models y = 2061x + 47,100
2010 is 70 years after 1940, so x = 70.
y = 2061(70) + 47,100
= $191,370
The median home price in New Jersey will be approximately $191,370.

12. 2.5 Using Linear Models
Assignment: p.96-97(#8,12bc,14bc,15-17) For #12 & 14, use these equations. 12.) y = x – 4,066, x = year (NOT # of years since 2000) 14.) y = x –

13. 2.6 Families of Functions
A parent function is the basic starting graph. A transformation is a change to the parent graph. Transformations can be translations or shifts of the graph up or down or left or right.

14. Examples of transformations
2.6 Families of Functions
Examples of transformations

15. 2.6 Families of Functions TRANSLATION UP or DOWN Begin with y = f(x).
To shift that graph up or down c units, we will write it
y = f(x) + c.
y = f(x) + 3
y = f(x) – 5

16. 2.6 Families of Functions TRANSLATION LEFT OR RIGHT
Begin with y = f(x).
To shift that graph left or right c units, we will write it
y = f(x + c) or y = f(x − c).
y = f(x + 4)
y = f(x − 6)

17. 2.6 Families of Functions
Given the graph of y = f(x), graph y = f(x)
+ 4
+ 4

18. 2.6 Families of Functions
Given the graph of y = f(x), graph y = f(x)
– 3
– 3

19. 2.6 Families of Functions
Given the graph of y = f(x), graph y = f(x ).
+ 4
+ 4

20. 2.6 Families of Functions
Given the graph of y = f(x), graph y = f(x ).
– 3
– 3

21. 2.6 Families of Functions Now, if y = f(x), graph y = f(x ) . – 2 – 2
+ 1
+ 1

22. 2.6 Families of Functions
Assignment: Worksheet (2.6) Translations

23. 2.6 Families of Functions
ANSWERS TO WORKSHEET 1.
f(x + 5)

24. 2.6 Families of Functions
ANSWERS TO WORKSHEET 2.
f(x) – 3

25. 2.6 Families of Functions
ANSWERS TO WORKSHEET 3.
f(x) + 3

26. 2.6 Families of Functions
ANSWERS TO WORKSHEET 4.
f(x ‒ 1) + 2

27. 2.6 Families of Functions
ANSWERS TO WORKSHEET 5.
f(x + 3) ‒ 4

28. 2.6 Families of Functions
ANSWERS TO WORKSHEET 6.
f(x ‒ 5) ‒ 3

29. 2.6 Families of Functions More Transformations: Reflection
f(−x) is a flip of f(x) over the y-axis.
− f(x) is a flip of f(x) over the x-axis.

30. 2.6 Families of Functions More Transformations (continued): Stretch
a∙f(x) is a vertical stretch by a factor of a; a > 1
Compression
a∙f(x) is a vertical compression by a factor of a; 0 < a < 1

31. 2.6 Families of Functions
Given y = f(x), graph y = f(– x).

32. 2.6 Families of Functions
Given y = f(x), graph y = – f(x). 32

33. 2.6 Families of Functions
Given y = f(x), graph y = 2f(x).

34. 2.6 Families of Functions
Given y = f(x), graph y = – 3f(x).

35. 2.6 Families of Functions
Given y = f(x), graph y = ½ f(– x).

36. 2.6 Families of Functions
Given y = f(x), graph y = – 2f(x) + 3.

37. 2.6 Families of Functions
Assignment: Worksheet (2.6 Enrichment)

38. 2.6 Families of Functions
ANSWERS ENRICHMENT WORKSHEET 4.
y = 2f(x)

39. 2.6 Families of Functions
ANSWERS ENRICHMENT WORKSHEET 5.
y = f(x) – 1

40. 2.6 Families of Functions
ANSWERS ENRICHMENT WORKSHEET 6.
y = f(x + 4)

41. 2.6 Families of Functions ANSWERS ENRICHMENT WORKSHEET 7.
y = 2f(x + 4) – 1

42. 2.6 Families of Functions
ANSWERS ENRICHMENT WORKSHEET 8.
y = f(x – 2)

43. 2.6 Families of Functions ANSWERS ENRICHMENT WORKSHEET 9.
y = – 2f(x) + 1

44. 2.6 Families of Functions ANSWERS ENRICHMENT WORKSHEET 10.
y = f(x + 3) – 4

45. 2.7 Absolute Value Graphs & Graphs
Graph f(x) = |x|. x y ‒2 ‒1 1 2 2 1 1 2

46. 2.7 Absolute Value Graphs & Graphs
Use the previous absolute value graph to answer the questions. What is the vertex? What are the slopes of the rays? What way does the graph open? What is the equation of the axis of symmetry?
(0,0) +1
and – 1
Up!
x = 0

47. 2.7 Absolute Value Graphs & Graphs
VERTEX FORM OF AN ABSOLUTE VALUE GRAPH

48. 2.7 Absolute Value Graphs & Graphs
The absolute value graph shifts UP if you see + k after the absolute value. The absolute value graph shifts DOWN if you see − k after the absolute value.

49. 2.7 Absolute Value Graphs & Graphs
Graph f(x) = |x| + 5 .
Shift the graph of f(x) = |x|
UP 5 units!!!
Remember to keep the slopes of the rays +1 and – 1!!!

50. 2.7 Absolute Value Graphs & Graphs
Use the previous absolute value graph to answer the questions. What is the vertex? What are the slopes of the rays? What way does the graph open? What is the equation of the axis of symmetry?
(0,5) +1
and – 1
Up!
x = 0

51. 2.7 Absolute Value Graphs & Graphs
The absolute value graph shifts LEFT h units if you see |x + h| in the equation. The absolute value graph shifts RIGHT h units if you see |x – h| in the equation.

52. 2.7 Absolute Value Graphs & Graphs
Graph f(x) = |x – 4| .
Shift the graph of f(x) = |x|
RIGHT 4 units!!!
Remember to keep the slopes of the rays +1 and – 1!!!

53. 2.7 Absolute Value Graphs & Graphs
Use the previous absolute value graph to answer the questions. What is the vertex? What are the slopes of the rays? What way does the graph open? What is the equation of the axis of symmetry?
(4,0) +1
and – 1
Up!
x = 4

54. 2.7 Absolute Value Graphs & Graphs
Graph f(x) = |x + 2| + 3 .
Shift the graph of f(x) = |x|
LEFT 2 units & UP 3 units!!!

55. 2.7 Absolute Value Graphs & Graphs
Use the previous absolute value graph to answer the questions. What is the vertex? What are the slopes of the rays? What way does the graph open? What is the equation of the axis of symmetry?
(– 2, 3) +1
and – 1
Up!
x = – 2

56. 2.7 Absolute Value Graphs & Graphs
Assignment: p.111(#8 – 16, 53) For #8 – 16, do not make a table of values. Shift the parent graph. Use a ruler!!!

57. 2.7 Absolute Value Graphs & Graphs
The absolute value graph REFLECTS over the x-axis if you see a negative in front of the absolute value.

58. 2.7 Absolute Value Graphs & Graphs
The absolute value graph is STRETCHED BY A FACTOR OF a if a > 1. The absolute value graph is COMPRESSED BY A FACTOR OF a if 0 < a < 1.

59. 2.7 Absolute Value Graphs & Graphs
Another way to graph absolute value graphs….. This method is especially useful when a is not 1.

60. 2.7 Absolute Value Graphs & Graphs
Use f(x) = ½ |x + 2| to find the following information. Vertex: Axis of symmetry: Direction of opening: Slopes of rays: List all transformations. Shift left 2. Compress by a factor of ½.
(– 2,0)
x = – 2 Up
+ ½
and – ½

61. 2.7 Absolute Value Graphs & Graphs
Graph f(x) = ½ |x + 2|.
1.) Plot the vertex.
V(– 2, 0)
2.) Rise and run to get both sides of the V that opens up.

62. 2.7 Absolute Value Graphs & Graphs
Use f(x) = – 2/3 |x + 3| + 4 to find the following information. Vertex: Axis of symmetry: Direction of opening: Slopes of rays: List all transformations.
(– 3, 4)
x = – 3
Down
± 2/3
Shift left 3,
shift up 4,
reflect over the x-axis,
and compress by a factor of 2/3.

63. 2.7 Absolute Value Graphs & Graphs
Graph f(x) = – 2/3 |x + 3| + 4 .
1.) Plot the vertex.
V(– 3, 4)
2.) Determine whether the V opens up or down.
This one: DOWN
3.) Rise and run to get both sides of the V that opens down.

64. 2.7 Absolute Value Graphs & Graphs
Use f(x) = – 3 |x – 5| – 3 to find the following information. Vertex: Axis of symmetry: Direction of opening: Slopes of rays: List all transformations.
(5, – 3)
x = 5
Down
± 3
Shift right 5,
shift down 3,
reflect over the x-axis,
and stretch by a factor of 3.

65. 2.7 Absolute Value Graphs & Graphs
Graph f(x) = – 3 |x – 5| – 3.
1.) Plot the vertex.
V(5, – 3)
2.) Determine whether the V opens up or down.
This one: DOWN
3.) Rise and run to get both sides of the V that opens down.

66. 2.7 Absolute Value Graphs & Graphs
Write an absolute value equation for the graph.

67. 2.7 Absolute Value Graphs & Graphs
Write an absolute value equation for the graph.

68. 2.7 Absolute Value Graphs & Graphs
Assignment: p.111(#17 – 30) For #23 – 28, find all of the information and then graph.

69. 2.6 Families of Functions
Given y = f(x), graph y = 2f(x).

70. 2.6 Families of Functions 70